Configuration Refinement; Freshness
To prepare for the semantics of threads and local variables, in this lesson we split the state cell into an environment and a store. The environment and the store will be similar to those in the definition of LAMBDA++ in Part 3 of the Tutorial. This configuration refinement will require us to change some of IMP's rules, namely those that used the state.
To split the state map, which binds program variables to values, into an environment mapping program variables to locations and a store mapping locations to values, we replace in the configuration declaration the cell
<state color="red"> .Map </state>
with two cells
<env color="LightSkyBlue"> .Map </env> <store color="red"> .Map </store>
Structurally speaking, this split of a cell into other cells is a major semantic change, which, unfortunately, requires us to revisit the existing rules that used the state cell. One could, of course, argue that we could have avoided this problem if we had followed from the very beginning the good-practice style to work with an environment and a store, instead of a monolithic state. While that is a valid argument, highlighting the fact that modularity is not only a feature of the framework alone, but one should also follow good practices to achieve it, it is also true that if all we wanted in Part 2 of the tutorial was to define IMP as is, then the split of the state in an environment and a store is unnecessary and not really justified.
The first rule which used a state cell is the lookup rule:
rule <k> X:Id => I ...</k> <state>... X |-> I ...</state>
We modify it as follows:
rule <k> X:Id => I ...</k> <env>... X |-> N ...</env> <store>... N |-> I ...</store>
So we first match the location
X in the environment, then the value
I at location
N in the store, and finally we rewrite
I into the
computation. This rule also shows an instance of a more complex
multiset matching, where two variables (
N) are matched each twice.
The assignment rule is modified quite similarly.
The variable declaration rule is trickier, though, because we need to allocate a fresh location in the store and bind the newly declared variable to it. This is quite similar to the way we allocated space for variables in the environment-based definition of LAMBDA++ in Part 3 of the tutorial.
rule <k> int (X,Xs => Xs); ...</k> <env> Rho => Rho[X <- !N:Int] </env> <store>... .Map => !N |-> 0 ...</store>
Note the use of the fresh (
!N) variable notation above. Recall from
the LAMBDA++ tutorial that each time the rule with fresh (
!) variables is
applied, fresh elements of corresponding sorts are generated for the fresh
variables, distinct from all the previously generated elements; also, we
cannot and should not assume anything about the particular element that is
being generated, except that it is different from the previous ones.
sum.imp to see how the fresh locations have been
generated and used. There were two fresh locations needed, for the two
variables. Note also that a cell holding the counter has been added to the
In the next lesson we will add the semantics of variable increment, and see how that yields non-deterministic behaviors in programs and how to explore those behaviors using the K tool.